Definition df-ms 15022

Description: Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)

Assertion

Ref	Expression
df-ms	⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}

Detailed syntax breakdown of Definition df-ms

Step	Hyp	Ref	Expression
1		cms 15019	. 2 class MetSp
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1394	. . . . . 6 class 𝑓
4		cds 13127	. . . . . 6 class dist
5	3, 4	cfv 5318	. . . . 5 class (dist‘𝑓)
6		cbs 13040	. . . . . . 7 class Base
7	3, 6	cfv 5318	. . . . . 6 class (Base‘𝑓)
8	7, 7	cxp 4717	. . . . 5 class ((Base‘𝑓) × (Base‘𝑓))
9	5, 8	cres 4721	. . . 4 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
10		cmet 14509	. . . . 5 class Met
11	7, 10	cfv 5318	. . . 4 class (Met‘(Base‘𝑓))
12	9, 11	wcel 2200	. . 3 wff ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))
13		cxms 15018	. . 3 class ∞MetSp
14	12, 2, 13	crab 2512	. 2 class {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
15	1, 14	wceq 1395	1 wff MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}

This definition is referenced by:  isms  15135